How do I make an appointment when nobody answers the phone. Tried two days in a row to make an appointment to take the written exam for my permit and there is a busy tone immediately

I ran into this to it's ridiculous. You have to go to skiptheline.ncdot.gov They will update you on all of the DMV's in the state that have an appointment available in the next two weeks. You have to keep checking back everyday until your local DMV has an appointment. It won't show up if there's nothing within the next two weeks. Once you finally make it you have to be within 15 minutes of your appointment time or they will make you reschedule. For April they only served at half of the capacity of the previous April. So it's more difficult than the regular rigmarole. However once you're there you're in and out pretty quick.

They have their phones directed to Raleigh. You can go on the NCDMV website and set up an appointment.